Suppose that a discrete random variable X has the following probability mass function, X 1 f(x) // 5 116 9 Find the moment generating function.
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- The following figure shows probability distributions, namely, Expon1 and Expon2, of inter-arrival times for phone calls at two different call centers. Note that the inter-arrival times in each call center are exponentially distributed, with rates Lambda1 and Lambda2, respectively. Expon(1) Expon(2) Based on the figure, which statement is true? Lambda 1 > Lambda 2 O Lambda 2> Lambda 1 O Lambda 1= Lambda 2 Not enough informationSuppose that Y1 is a binomial random variable with four trials and success probability 0.7 and that Y2 is an independent binomial random variable with three trials and success probability 0.5. Let W = Y1 + Y2. W does not have a binomial distribution. Find the probability mass function for W. [HINT: P(W = 0) = P(Y1 = 0, Y2 = 0); P(W = 1) = P(Y1 = 1, Y2 = 0) + P(Y1 = 0, Y2 = 1); etc.] (Round your answers to four decimal places.)The element titanium has five stable occurring isotopes, differing from each other in the number of neutrons an atom contains. If X is the number of neutrons in a randomly chosen titanium atom, the probability mass function of X is given as follows: 24 25 26 27 28 х 0.0541 p(x) 0.0518 0.0825 0.0744 0.7372 Find Hx- a.
- Let X be a random variable with pdf f(x) = 0.2 €-0.2x 0 if x > 0, otherwise. (a) Find the moment generating function of X, remembering to state the range for which it holds. (b) Hence find the expectation and variance of X.Assume that 1 out of every 5 adults in a local community is unemployed. Approximatethe probability that in a random sample of 100 adults from the community, more than 25are unemployed. Express your final answer in terms of the standard normal cumulativedistribution function Φ.Show that al(w) is a random variable, where EEF and a is a constant. Calculate E[X] using the basic definition of an expectation as done in class.
- A bank operates both a drive-up facility and a walk-up window. On a randomly selected day, let X be the proportion of time that the drive-up facility is in use (at least one customer is being served or waiting to be served) and Y be the proportion of time that the walk-up window is in use. The joint PDF is ... fxy (x, y) = 5 (x + y²) • What is the probability that neither window is busy more than one-quarter of the time? • What is f(X)? • What is fy(y)?Verify that the following function is a probability mass function, and determine the requested probabilities. Answer rounded-off to 4 decimal places f(x)=(125/31)(1/5)* P(X>1)=_ 7 x={1,2,3}For E (0,1) let Xp be a Geometric random variable with parameter p. (a) Find a value of p so that P(Xp> 2.5) = 9. (b) Let An be the event that Xp is even. Determine P(A,) in terms of p. (c) Suppose Y, is a random variable which is equal to the remainder after integer division of X, by 3. Let p = i, and determine the conditional probability mass function of conditioned on the event Yı = 1.
- Suppose that the moment-generating function of a random variable X is given by Mx(1) = +" +" + 4 3t 2 4 5t 15 Find the probability mass function of X.Suppose X is a discrete random variable which only takes on positive integer values. For the cumulative distribution function associated to X the following values are known: F(23) 0.34 F(29) = =0.38 F(34) 0.42 F(39) 0.47 F(44) = 0.52 F(49) 0.55 F(56) = 0.61 = Determine Pr[29Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.33 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. (a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.) 0 1 2 3 4 5 6 7 8 P(X=x) 0 1 2 3 4 5 6 7 0.1350 0.1330 0.2322 0.1965 0.2633 ✓ ✔ (b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.) F(x) 0.1350 0.268 0.5002 0.6967 x
